#### Volume 18, issue 5 (2018)

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A refinement of Betti numbers and homology in the presence of a continuous function, II: The case of an angle-valued map

### Dan Burghelea

Algebraic & Geometric Topology 18 (2018) 3037–3087
##### Abstract

For $f:X\to {\mathbb{S}}^{1}$ a continuous angle-valued map defined on a compact ANR $X\phantom{\rule{0.3em}{0ex}}$, $\kappa$ a field and any integer $r\ge 0$, one proposes a refinement ${\delta }_{r}^{f}$ of the Novikov–Betti numbers of the pair $\left(X,{\xi }_{f}\right)$ and a refinement ${\stackrel{̂}{\delta }}_{r}^{f}$ of the Novikov homology of $\left(X,{\xi }_{f}\right)$, where ${\xi }_{f}$ denotes the integral degree one cohomology class represented by $f$. The refinement ${\delta }_{r}^{f}$ is a configuration of points, with multiplicity located in ${ℝ}^{2}∕ℤ$ identified to $ℂ\setminus 0$, whose total cardinality is the Novikov–Betti number of the pair. The refinement ${\stackrel{̂}{\delta }}_{r}^{f}$ is a configuration of submodules of the Novikov homology whose direct sum is isomorphic to the Novikov homology and with the same support as of ${\delta }_{r}^{f}\phantom{\rule{0.3em}{0ex}}$. When $\kappa =ℂ$, the configuration ${\stackrel{̂}{\delta }}_{r}^{f}$ is convertible into a configuration of mutually orthogonal closed Hilbert submodules of the ${L}_{2}$–homology of the infinite cyclic cover of $X$ defined by $f$, which is an ${L}^{\infty }\left({\mathbb{S}}^{1}\right)$–Hilbert module. One discusses the properties of these configurations, namely robustness with respect to continuous perturbation of the angle-values map and the Poincaré duality and one derives some computational applications in topology. The main results parallel the results for the case of real-valued map but with Novikov homology and Novikov–Betti numbers replacing standard homology and standard Betti numbers.

##### Keywords
Novikov–Betti numbers, angle-valued maps, barcodes
##### Mathematical Subject Classification 2010
Primary: 46M20, 55N35, 57R19